Similar to the empirical dataset, there are no orientational order changes during the liquid-vapor phase transition in our model. It is interesting to explore such a magnetization phase transition, similar to the celebrated Berezinskii–Kosterlitz–Thouless transition observed in the classical \(XY\) -model [1]}, [2]} and existing social force models [3]}, [4]}, [5]}, [6]}. Indeed, we observe a similar phase transition that occurs at very high-density values around \(\bar{\rho }\gg 1\) , about one order of magnitude larger than that observed empirically. Nevertheless, the magnetization order in our model shows a striped pattern instead of a ferromagnetic phase (see Inset of Fig. 4c). This reflects the fact that our coupling function \(J(r)\)  is ferromagnetic in short distances and antiferromagnetic at longer distances. Figure 4c plots the stripped order parameter \(m_s\)  and susceptibility \(\chi _s\)  as functions of temperature (see SM Section D3 for definitions), providing clear evidence of such a phase transition. Note that this orientational phase transition occurs independently of the vapor-liquid phase transition, implying that the underlying physics of our model Eq. (REF ) is distinct from those of existing models [3]}, [8]}.
